They're not as easy to build and use as you might have us believe, but we do use Wankel engines. In automotive applications, Mazda favour them for sports cars due to their high power to weight ratio, torque curves, and high revving nature.

The main reasons not to use them are their high fuel consumption, high oil consumption, and the more difficult engineering required in their manufacture (namely problems with sealing dissimilar metals at elevated temperatures, which partially causes the high oil consumption). The thermal problems are excarbated by the fact that the combustion chamber is not cooled by the incoming charge as would be with a reciprocating engine. Very high HC emissions are also a large problem, presumably due to the combustion chamber geometry intrinsically preventing complete combustion. Obviously the HC problem can be overcome by richening the mixture, allowing complete combustion to occur in the exhaust manifold following expansion, but with today's fuel costs this is just daft. Another major problem is that with some designs, carbon deposits can accumulate between the rotor seal and the crankcase, causing the engine to sieze if it's not allowed to fully warm through.

It should be noted that Mazda have managed to successfully address many of these issues, however, fuel consumption is still pretty hairy.

I'm still waiting to see the design for your non-reciprocating piston engine!